# Integral of Arcsin[x]

Gold Member
MHB
I just can't see it. I would think that it would involve some form of trig substitution, but I'm just drawing a blank. I'll do the work if someone can please give me a nice little hint.

I know that $$\int \arcsin{(x)}dx = \sqrt{1-x^2}+ x\arcsin{(x}) + C$$ from my calculator and mathematica.

Staff Emeritus
Gold Member
If only you were working with its derivative! I bet you know how to integrate something that looks like its derivative!

Gold Member
MHB
Sure... I'd draw a nice triangle and do trig substitution if it was $$\int \frac{dx}{\sqrt{1-x^2}}$$

Hmmm...I'll think out loud here. If the integral has a square-root and is in the form of $c^2-x^2$ then x is one leg, c is the hypotenuse, and the other leg is the previously mentioned radical.

Is trig substitution with right triangles on the right track? Since the hypotenuse is $\sqrt{a^2+b^2}$, it seems that one leg might need to be $\sqrt{\arcsin{(x)}}$ Somehow I don't think I'm on track.

amcavoy
Use integration by parts. Remember to let u=arcsin(x) and v=x.

u sub: InverseLogAlgebraicTrigExp

Staff Emeritus
Gold Member
Sheesh, just give him the answer, why don't ya? :grumpy:

Gold Member
MHB
apmcavoy said:
Use integration by parts. Remember to let u=arcsin(x) and v=x.

u sub: InverseLogAlgebraicTrigExp

Ah, thank you! It's so simple.

And thank you Hurkyl as well. I still have lots to learn.

amcavoy
I apologize Hurkyl tongos
well, you know the integral of sinx with limits. Now arcsin x will be the limits, and you can make a rectangle.

Or you could just take the derivative of the right hand side and go "ta da!" and that's proof enough for me.

professorlucky
hint

equate the arc sine to another variable e.g y.making it a sine fxn.e.g the arc sine of 0.5=30,while sine30=0.5.this will simplify the integral and further substitution will conclude it

Homework Helper
Just for the fun of it ...

The sum of the integrals

$$\int \sin^{-1} x dx + \int \sin y dy$$

is just the area of the bounding rectangle: $x \times \sin^{-1} x$

Since $$\int \sin y dy = -\cos y + C$$ and $$\cos y = \cos \sin^{-1} x = \sqrt {1-x^2}$$ it follows that
$$\int \sin^{-1} x dx = \sqrt {1-x^2} + x \sin^{-1} x + C$$

FlashStorm
Integration by parts?

Hey, I am really sorry to arise dead threads from the past (which i have seen though google) but somehting really weird happened me when I tried to use integration by parts on arcsinx.

let me show you:
S(arcsinx)= {v'(x)=1} {u(x)=arcsinx}

xarcsinx-S(x*d(arcsinx))=

xarcsinx-S(x/(1-x^2)^0.5= {u(x)=x v'(x)=arcsinx}

xarcsinx-xarcsinx+S(arcsin)dx

==> S(arcsinx)=S(arcsinx)

:\
I know I have done something really stupid here, but please be easy on me since I started studying Integrals only three days ago.

In the second time I used integration by parts, do I miss something , is there another efficient choice of v and u?

Aviv

p.s: I will edit it better to use normal math signs once I figure out how.

Homework Helper
For the second time you integrate by parts, swap your choices.

FlashStorm
Im sorry, Tried it also and all i got is:

xarcsine x -(x^2/2)(1/(Sqrt(1-x^2)))-(x/4)(sqrt(1/(1-x^2))+1/4(arcsinx)

this isn't going anywhere :(

Homework Helper
I just can't see it. I would think that it would involve some form of trig substitution, but I'm just drawing a blank. I'll do the work if someone can please give me a nice little hint.

I know that $$\int \arcsin{(x)}dx = \sqrt{1-x^2}+ x\arcsin{(x}) + C$$ from my calculator and mathematica.

Make $x=\sin t$. Then apply part integration on the resulting integral. It's just a way to avoid the simple solution of part integrating directly.

Staff Emeritus
Gold Member
After the first integration by parts, I would use a substitution

FlashStorm
solve it officially. Fixed during some major mistakes I had about dev' and stuff.

did it without subtition, only using integration by parts.

if you are interested what I did then you are welcome to tell me to write my solution.

Thanks guys :)
gg

josiahsavino
use separation by parts

u=arcsinx
du=dx/(1-x^2)^1/2

dv=dx
v=x

uv-ingegral vdu = xarsinx-integral x/(1-x^2)^1/2
use u substitution with u = 1-x^2 so du = -2x
than you get

xarsinx + (1-x^2)^1/2 + c

Liqorice